Sample desctiptives

Correlation matrix

## 
## 
## Table 1 
## 
## Means, standard deviations, and correlations with confidence intervals
##  
## 
##   Variable  M     SD   1            2           
##   1. PreMP  4.60  2.65                          
##                                                 
##   2. PreMA  13.58 6.00 -.25**                   
##                        [-.30, -.20]             
##                                                 
##   3. PreMSE 3.50  0.96 .36**        -.58**      
##                        [.31, .41]   [-.61, -.54]
##                                                 
## 
## Note. M and SD are used to represent mean and standard deviation, respectively.
## Values in square brackets indicate the 95% confidence interval.
## The confidence interval is a plausible range of population correlations 
## that could have caused the sample correlation (Cumming, 2014).
##  * indicates p < .05. ** indicates p < .01.
##

Sample demographics

Characteristic N = 1,6491
Gender
    Female 782 (47%)
    Male 867 (53%)
    Unknown 0 (0%)
race_ethnicity
    American Indian/Alaska Native 10 (0.6%)
    Asian 406 (25%)
    Black/African American 82 (5.0%)
    Hispanic/Latino 268 (17%)
    Native Hawaiian or Other Pacific Islander 0 (0%)
    Two or more races 52 (3.2%)
    White 806 (50%)
    Unknown 25
IEP
    0 1,461 (89%)
    1 188 (11%)
EIP
    0 1,535 (93%)
    1 114 (6.9%)
GIFTED
    0 1,387 (84%)
    1 262 (16%)
ESOL
    0 1,485 (90%)
    1 164 (9.9%)
PreMP 4 (3, 7)
    Unknown 357
PreMA 14 (9, 18)
    Unknown 443
PreMSE 3.60 (3.00, 4.20)
    Unknown 448
1 n (%); Median (Q1, Q3)

Analysis

Clustering

Choosing number of clusters

Dropping NAs and Z-scoring

### --- Delete cases with NAs in Pre Math anxiety or Pre Math performance
FH2T <- FH2T %>% drop_na(PreMA)
FH2T <- FH2T %>% drop_na(PreMP)

# Z-scoring MP and MA
FH2T$PreMP_z <- 
  (FH2T$PreMP - mean(FH2T$PreMP))/sd(FH2T$PreMP)
FH2T$PreMA_z <- 
  (FH2T$PreMA - mean(FH2T$PreMA))/sd(FH2T$PreMA)

# Creating new dataframes for PRE-levels clustering based on scaled variables
PRE_z <- FH2T %>% as.data.frame() %>%
  dplyr::select(PreMA_z, PreMP_z)

Elbow method

fviz_nbclust(PRE_z, kmeans, method = "wss") +
  geom_vline(xintercept = 4, linetype = 2)+
  labs(subtitle = "Elbow method")

Silhouette scores

# Range of cluster numbers to test
silhouette_scores <- numeric(10)

# Loop through different numbers of clusters
for (k in 2:10) {
  set.seed(123)  # For reproducibility
  kmeans_result <- kmeans(PRE_z, centers = k)
  sil <- silhouette(kmeans_result$cluster, dist(PRE_z))
  silhouette_scores[k] <- mean(sil[, 3])  # Average Silhouette score for this k
}

# Find the number of clusters with the highest average Silhouette score
best_k <- which.max(silhouette_scores)
cat("The optimal number of clusters is", best_k, "with an average Silhouette score of", silhouette_scores[best_k], "\n")
## The optimal number of clusters is 4 with an average Silhouette score of 0.3927846
# Plot the Silhouette scores for each number of clusters
plot(2:10, silhouette_scores[2:10], type = "b",
     xlab = "Number of Clusters", ylab = "Average Silhouette Score",
     main = "Silhouette Score for Different Numbers of Clusters")

Clustering with 4 centers

### --- Applying k-means clustering
set.seed(20)
pre_cluster <- kmeans(PRE_z, centers = 4, nstart = 25) # put the optimal number of clusters in "centers"
print(pre_cluster)   
## K-means clustering with 4 clusters of sizes 227, 261, 375, 343
## 
## Cluster means:
##      PreMA_z    PreMP_z
## 1 -1.0654934  1.3278878
## 2  0.5822366  0.7544672
## 3 -0.6699864 -0.6206387
## 4  0.9946010 -0.7743643
## 
## Clustering vector:
##    [1] 1 1 4 2 3 4 3 3 3 1 3 1 3 3 3 4 4 4 4 4 3 2 3 4 3 2 3 3 3 3 4 3 3 4 3 3 3
##   [38] 3 3 3 3 3 3 4 2 3 4 4 4 4 4 3 4 1 2 2 2 4 3 2 2 2 4 4 3 4 4 2 2 4 3 3 4 4
##   [75] 2 2 2 2 2 3 4 3 4 3 3 4 4 3 3 3 3 4 4 4 4 3 3 3 2 4 4 2 4 4 4 3 4 4 4 4 4
##  [112] 2 4 4 4 2 4 3 4 3 4 4 2 4 3 2 4 4 4 4 4 4 4 4 2 4 4 2 4 4 3 2 4 4 2 2 3 2
##  [149] 2 1 3 3 4 4 3 4 3 2 4 3 3 4 2 4 4 4 1 3 3 4 2 4 3 3 2 4 1 3 3 4 4 3 4 3 3
##  [186] 4 3 3 2 3 3 4 4 3 4 4 1 2 4 4 3 4 3 3 4 2 2 2 2 2 1 4 1 1 4 1 3 2 4 1 1 2
##  [223] 2 3 2 1 3 2 1 3 3 3 4 3 3 4 2 2 1 2 1 4 2 2 2 1 3 3 3 1 2 4 4 4 4 4 3 3 4
##  [260] 4 3 4 4 4 3 3 3 3 4 3 4 4 3 4 4 3 3 3 3 3 3 4 3 3 3 4 3 4 3 3 4 4 4 3 4 3
##  [297] 3 3 4 4 4 3 4 4 2 3 3 4 4 1 3 4 4 4 4 4 3 3 3 4 4 4 4 4 4 3 3 4 3 3 3 3 3
##  [334] 3 4 3 3 2 3 3 4 3 3 4 4 4 3 4 2 3 4 4 4 3 4 2 3 3 3 2 4 4 3 3 3 3 3 4 4 3
##  [371] 3 3 4 3 3 3 3 3 4 3 3 4 3 4 2 4 4 3 4 3 4 4 4 4 4 3 3 3 3 3 3 4 4 3 4 3 4
##  [408] 4 3 4 4 3 4 4 4 4 3 4 4 1 4 3 4 4 1 4 4 1 3 2 1 2 4 3 3 2 3 4 3 2 2 2 2 2
##  [445] 3 2 3 3 1 3 2 4 2 2 2 2 3 1 2 3 4 2 2 3 3 3 2 4 3 4 3 4 4 4 3 4 4 3 4 4 3
##  [482] 3 4 3 3 3 3 2 4 4 4 2 4 3 1 1 2 3 3 3 1 1 3 4 3 4 2 3 4 3 1 4 3 3 4 3 3 4
##  [519] 2 3 4 1 3 4 4 2 3 3 3 3 3 2 1 3 4 4 3 3 3 4 2 1 3 3 3 3 4 4 4 2 1 2 1 2 3
##  [556] 4 1 2 1 1 3 3 3 4 4 3 4 4 4 3 3 2 3 1 3 3 2 3 3 2 4 4 4 4 3 3 4 3 3 3 3 2
##  [593] 3 2 1 1 3 2 2 2 2 3 2 3 1 3 4 3 3 3 2 4 2 3 2 1 1 4 2 4 1 2 4 4 3 2 3 4 1
##  [630] 4 3 1 4 4 4 4 2 2 4 2 1 4 1 4 2 3 2 1 4 2 4 3 4 4 2 3 3 4 4 4 1 4 4 4 4 3
##  [667] 3 1 1 3 3 3 1 4 3 4 1 3 3 1 4 2 3 2 2 1 4 1 4 1 3 3 3 3 3 4 3 4 3 2 4 3 3
##  [704] 4 2 3 1 2 1 2 2 1 2 1 1 1 1 2 2 2 1 3 4 2 3 3 2 3 4 4 3 3 3 3 3 4 4 4 1 4
##  [741] 3 4 2 3 4 3 3 4 3 4 2 4 4 3 3 3 4 3 1 4 3 3 3 4 3 3 3 1 3 2 3 4 4 4 3 4 4
##  [778] 4 2 3 3 4 4 1 2 4 3 3 3 4 4 4 3 4 3 2 3 3 4 4 4 2 3 4 3 1 3 4 4 2 1 1 1 1
##  [815] 1 1 1 1 1 2 1 1 1 2 2 2 1 2 1 1 4 3 2 3 2 1 2 2 1 3 1 2 1 4 1 2 4 1 2 1 1
##  [852] 1 1 2 1 1 2 1 1 2 1 2 3 2 2 1 3 1 1 1 1 1 2 2 2 2 2 2 3 1 1 1 2 3 4 3 2 3
##  [889] 2 2 3 1 2 2 1 2 4 4 4 2 2 2 2 1 2 3 3 3 2 1 2 1 2 1 1 2 1 1 1 2 1 2 1 2 2
##  [926] 1 2 1 1 1 3 2 3 1 3 2 1 4 2 1 1 2 1 1 1 2 1 2 1 1 2 1 1 1 1 1 1 2 2 1 2 1
##  [963] 1 4 1 1 1 2 2 3 3 3 4 3 1 3 1 3 1 2 1 4 3 1 2 1 1 1 1 1 1 1 1 2 4 4 2 3 3
## [1000] 4 3 4 3 4 3 3 3 2 4 2 4 2 3 4 2 2 2 4 4 3 2 4 4 4 3 2 4 3 3 4 2 2 4 4 2 2
## [1037] 4 1 4 1 4 2 2 1 2 1 2 2 1 2 2 1 2 1 1 3 3 1 2 1 1 2 1 1 1 2 1 2 1 1 3 3 2
## [1074] 3 2 2 1 3 2 3 4 3 1 3 1 2 2 1 3 1 2 1 2 2 1 2 2 3 2 4 4 1 2 2 4 2 1 1 1 1
## [1111] 4 4 2 3 2 4 4 3 3 1 2 2 2 1 1 2 1 2 2 2 2 1 1 1 1 1 1 2 1 1 1 2 2 1 2 2 3
## [1148] 1 2 1 4 3 4 4 4 3 4 2 4 3 3 4 3 1 1 1 3 1 1 1 3 3 3 2 1 2 3 3 2 1 1 2 1 4
## [1185] 3 4 3 3 3 2 4 2 3 2 4 1 3 1 1 1 1 1 2 2 2 1
## 
## Within cluster sum of squares by cluster:
## [1] 119.6462 156.7847 210.1404 170.6491
##  (between_SS / total_SS =  72.7 %)
## 
## Available components:
## 
## [1] "cluster"      "centers"      "totss"        "withinss"     "tot.withinss"
## [6] "betweenss"    "size"         "iter"         "ifault"
# Save the cluster number in the dataset as column 'cluster_results'
FH2T$pre_cluster_results <- as.factor(pre_cluster$cluster)

Visualizing clusters

# Calculate centroids from your K-means result
centroids <- as.data.frame(pre_cluster$centers)

cluster_colors <- c("#E69F00", "#56B4E9", "#009E73", "#F0E442")

# Visualize the data with ggplot
library(ggplot2)
ggplot(FH2T, aes(PreMA_z, PreMP_z)) +
  geom_jitter(aes(color = factor(pre_cluster_groups))) +
  geom_point(data = centroids, aes(x = PreMA_z, y = PreMP_z), 
             color = "black", size = 4, shape = 8) +  # Red stars for centroids
  scale_color_manual(values = cluster_colors) +
  labs(color = "Cluster", x = "Math Anxiety Score", y = "Math Test Score") +
  theme_minimal()

Clusters’ demographics

Characteristic lMP_hMA
N = 3431		 lMP_lMA
N = 3751		 hMP_lMA
N = 2271		 hMP_hMA
N = 2611
Gender



    Female 198 (58%) 131 (35%) 81 (36%) 153 (59%)
    Male 145 (42%) 244 (65%) 146 (64%) 108 (41%)
    Unknown 0 (0%) 0 (0%) 0 (0%) 0 (0%)
race_ethnicity



    American Indian/Alaska Native 3 (0.9%) 1 (0.3%) 2 (0.9%) 2 (0.8%)
    Asian 21 (6.3%) 38 (10%) 132 (58%) 91 (35%)
    Black/African American 18 (5.4%) 18 (4.8%) 2 (0.9%) 9 (3.5%)
    Hispanic/Latino 80 (24%) 72 (19%) 14 (6.2%) 31 (12%)
    Native Hawaiian or Other Pacific Islander 0 (0%) 0 (0%) 0 (0%) 0 (0%)
    Two or more races 7 (2.1%) 13 (3.5%) 4 (1.8%) 10 (3.9%)
    White 207 (62%) 230 (62%) 72 (32%) 115 (45%)
    Unknown 7 3 1 3
IEP



    0 298 (87%) 312 (83%) 215 (95%) 243 (93%)
    1 45 (13%) 63 (17%) 12 (5.3%) 18 (6.9%)
EIP



    0 312 (91%) 336 (90%) 222 (98%) 255 (98%)
    1 31 (9.0%) 39 (10%) 5 (2.2%) 6 (2.3%)
GIFTED



    0 328 (96%) 353 (94%) 124 (55%) 197 (75%)
    1 15 (4.4%) 22 (5.9%) 103 (45%) 64 (25%)
ESOL



    0 290 (85%) 326 (87%) 220 (97%) 240 (92%)
    1 53 (15%) 49 (13%) 7 (3.1%) 21 (8.0%)
PreMP 3 (2, 4) 3 (2, 4) 8 (7, 9) 6 (6, 8)
PreMA 19 (17, 22) 10 (7, 12) 7 (5, 10) 17 (15, 19)
PreMSE 2.80 (2.20, 3.60) 3.60 (3.00, 4.20) 4.40 (3.80, 4.80) 3.40 (3.00, 4.00)
    Unknown 0 3 1 1
1 n (%); Median (Q1, Q3)

Comparison by MP

# Checking normality - normally distributed
FH2T %>%
  group_by(pre_cluster_groups) %>%
  summarise(shapiro_statistic = shapiro.test(PreMP)$statistic,
            p.value = shapiro.test(PreMP)$p.value)
# Checking homogeneity of variance - not normally distributed
leveneTest(PreMP ~ pre_cluster_groups, data = FH2T)
bartlett.test(PreMP ~ pre_cluster_groups, data = FH2T)
## 
##  Bartlett test of homogeneity of variances
## 
## data:  PreMP by pre_cluster_groups
## Bartlett's K-squared = 12.749, df = 3, p-value = 0.005212
## MP comparison via Dunn test, as variances not normally distributed
dunn.test(FH2T$PreMP, g=FH2T$pre_cluster_groups, method='bonferroni')
##   Kruskal-Wallis rank sum test
## 
## data: x and group
## Kruskal-Wallis chi-squared = 884.7216, df = 3, p-value = 0
## 
## 
##                            Comparison of x by group                            
##                                  (Bonferroni)                                  
## Col Mean-|
## Row Mean |    hMP_hMA    hMP_lMA    lMP_hMA
## ---------+---------------------------------
##  hMP_lMA |  -4.588898
##          |    0.0000*
##          |
##  lMP_hMA |   19.80806   23.88337
##          |    0.0000*    0.0000*
##          |
##  lMP_lMA |   17.72150   21.93966  -2.656532
##          |    0.0000*    0.0000*    0.0237*
## 
## alpha = 0.05
## Reject Ho if p <= alpha/2

Comparison by MA

# Checking normality - not normally distributed
FH2T %>%
  group_by(pre_cluster_groups) %>%
  summarise(shapiro_statistic = shapiro.test(PreMA)$statistic,
            p.value = shapiro.test(PreMA)$p.value)
# Checking homogeneity of variance - normally distributed
leveneTest(PreMA ~ pre_cluster_groups, data = FH2T)
bartlett.test(PreMA ~ pre_cluster_groups, data = FH2T)
## 
##  Bartlett test of homogeneity of variances
## 
## data:  PreMA by pre_cluster_groups
## Bartlett's K-squared = 0.43606, df = 3, p-value = 0.9327
## MA comparison via Dunn test, as data is not normally distributed
dunn.test(FH2T$PreMA, g=FH2T$pre_cluster_groups, method='bonferroni')
##   Kruskal-Wallis rank sum test
## 
## data: x and group
## Kruskal-Wallis chi-squared = 913.3897, df = 3, p-value = 0
## 
## 
##                            Comparison of x by group                            
##                                  (Bonferroni)                                  
## Col Mean-|
## Row Mean |    hMP_hMA    hMP_lMA    lMP_hMA
## ---------+---------------------------------
##  hMP_lMA |   18.75219
##          |    0.0000*
##          |
##  lMP_hMA |  -4.841687  -24.53880
##          |    0.0000*    0.0000*
##          |
##  lMP_lMA |   16.57079  -4.353373   23.20161
##          |    0.0000*    0.0000*    0.0000*
## 
## alpha = 0.05
## Reject Ho if p <= alpha/2

Vizualization of comparison by MP and MA (z-scored)

## Visualization for both
 
# Creating long format table
FH2T_data_long <- pivot_longer(FH2T,
                               cols = c('PreMP_z', 'PreMA_z'),
                               names_to = 'Variable',
                               values_to = 'Value')

# Specify levels for factor "Variable" (so MP goes first on the visualization)
FH2T_data_long$Variable <- factor(FH2T_data_long$Variable , levels=c("PreMP_z", "PreMA_z"))
 
 
# Create a boxplot for each variable with facets for clusters
ggplot(FH2T_data_long, aes(x = Variable , y = Value, fill = Variable)) +
  geom_boxplot() +
  labs(x = "Cluster", y = "Value") +
  facet_wrap(~ pre_cluster_groups_best, scales = "fixed") +
  scale_fill_manual(values = wes_colors) +
  theme_minimal()

MP distribution

MA distribution

Compare by MSE

Z-scoring, checking normality and comparing MSE

# Checking normality - not normally distributed
FH2T %>%
  group_by(pre_cluster_groups) %>%
  summarise(shapiro_statistic = shapiro.test(PreMSE)$statistic,
            p.value = shapiro.test(PreMSE)$p.value)
# Checking homogeneity of variance - not normally distributed
leveneTest(PreMSE ~ pre_cluster_groups, data = FH2T)
bartlett.test(PreMSE ~ pre_cluster_groups, data = FH2T)
## 
##  Bartlett test of homogeneity of variances
## 
## data:  PreMSE by pre_cluster_groups
## Bartlett's K-squared = 46.563, df = 3, p-value = 4.304e-10
## MA comparison via Dunn test, as data is not normally distributed
dunn.test(FH2T$PreMSE, g=FH2T$pre_cluster_groups, method='bonferroni')
##   Kruskal-Wallis rank sum test
## 
## data: x and group
## Kruskal-Wallis chi-squared = 316.2546, df = 3, p-value = 0
## 
## 
##                            Comparison of x by group                            
##                                  (Bonferroni)                                  
## Col Mean-|
## Row Mean |    hMP_hMA    hMP_lMA    lMP_hMA
## ---------+---------------------------------
##  hMP_lMA |  -10.22075
##          |    0.0000*
##          |
##  lMP_hMA |   6.906861   17.47843
##          |    0.0000*    0.0000*
##          |
##  lMP_lMA |  -2.744899   8.390480  -10.55111
##          |    0.0182*    0.0000*    0.0000*
## 
## alpha = 0.05
## Reject Ho if p <= alpha/2

Visualizing MSE distribution

## Warning: Removed 5 rows containing non-finite outside the scale range
## (`stat_density()`).

Visualizinf with MP and MA within groups

## Warning: Removed 5 rows containing non-finite outside the scale range
## (`stat_boxplot()`).

## 
## Call:
## lm(formula = PreMP_rank_z ~ PreMA_rank_z * PreMSE_rank_z, data = ranked_data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.32199 -0.75799  0.02009  0.73880  2.16092 
## 
## Coefficients:
##                            Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                -0.04051    0.03083  -1.314  0.18908    
## PreMA_rank_z               -0.05009    0.03224  -1.553  0.12058    
## PreMSE_rank_z               0.33013    0.03223  10.244  < 2e-16 ***
## PreMA_rank_z:PreMSE_rank_z -0.07349    0.02755  -2.668  0.00774 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9317 on 1202 degrees of freedom
## Multiple R-squared:  0.1341, Adjusted R-squared:  0.1319 
## F-statistic: 62.05 on 3 and 1202 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = PreMA_rank_z ~ PreMP_rank_z * PreMSE_rank_z, data = ranked_data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.23457 -0.63893 -0.05456  0.66925  2.25269 
## 
## Coefficients:
##                            Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                 0.02312    0.02554   0.905   0.3656    
## PreMP_rank_z               -0.04013    0.02569  -1.562   0.1185    
## PreMSE_rank_z              -0.53712    0.02565 -20.944   <2e-16 ***
## PreMP_rank_z:PreMSE_rank_z -0.06496    0.02490  -2.609   0.0092 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8317 on 1202 degrees of freedom
## Multiple R-squared:  0.3099, Adjusted R-squared:  0.3082 
## F-statistic:   180 on 3 and 1202 DF,  p-value: < 2.2e-16